- FindStat

Identifier

Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
St000386: Dyck paths ⟶ ℤ

Values

[1,0] => [1,0] => 0

[1,0,1,0] => [1,1,0,0] => 0

[1,1,0,0] => [1,0,1,0] => 0

[1,0,1,0,1,0] => [1,1,1,0,0,0] => 0

[1,0,1,1,0,0] => [1,0,1,1,0,0] => 0

[1,1,0,0,1,0] => [1,1,0,1,0,0] => 0

[1,1,0,1,0,0] => [1,1,0,0,1,0] => 1

[1,1,1,0,0,0] => [1,0,1,0,1,0] => 0

[1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => 0

[1,0,1,0,1,1,0,0] => [1,0,1,1,1,0,0,0] => 0

[1,0,1,1,0,0,1,0] => [1,1,1,0,1,0,0,0] => 0

[1,0,1,1,0,1,0,0] => [1,0,1,1,0,1,0,0] => 0

[1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,0,0] => 0

[1,1,0,0,1,0,1,0] => [1,1,0,1,1,0,0,0] => 0

[1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0] => 1

[1,1,0,1,0,0,1,0] => [1,1,0,1,0,1,0,0] => 0

[1,1,0,1,0,1,0,0] => [1,1,0,1,0,0,1,0] => 1

[1,1,0,1,1,0,0,0] => [1,1,0,0,1,0,1,0] => 1

[1,1,1,0,0,0,1,0] => [1,1,1,0,0,1,0,0] => 1

[1,1,1,0,0,1,0,0] => [1,1,1,0,0,0,1,0] => 1

[1,1,1,0,1,0,0,0] => [1,0,1,1,0,0,1,0] => 1

[1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => 0

[1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => 0

[1,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,0,0,0,0] => 0

[1,0,1,0,1,1,0,0,1,0] => [1,1,1,1,0,1,0,0,0,0] => 0

[1,0,1,0,1,1,0,1,0,0] => [1,0,1,1,1,0,1,0,0,0] => 0

[1,0,1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => 0

[1,0,1,1,0,0,1,0,1,0] => [1,1,1,0,1,1,0,0,0,0] => 0

[1,0,1,1,0,0,1,1,0,0] => [1,0,1,1,0,1,1,0,0,0] => 0

[1,0,1,1,0,1,0,0,1,0] => [1,1,1,0,1,0,1,0,0,0] => 0

[1,0,1,1,0,1,0,1,0,0] => [1,0,1,1,0,1,0,1,0,0] => 0

[1,0,1,1,0,1,1,0,0,0] => [1,0,1,1,0,0,1,1,0,0] => 1

[1,0,1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,1,0,0,0] => 1

[1,0,1,1,1,0,0,1,0,0] => [1,0,1,1,1,0,0,1,0,0] => 1

[1,0,1,1,1,0,1,0,0,0] => [1,0,1,0,1,1,0,1,0,0] => 0

[1,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => 0

[1,1,0,0,1,0,1,0,1,0] => [1,1,0,1,1,1,0,0,0,0] => 0

[1,1,0,0,1,0,1,1,0,0] => [1,1,0,0,1,1,1,0,0,0] => 1

[1,1,0,0,1,1,0,0,1,0] => [1,1,0,1,1,0,1,0,0,0] => 0

[1,1,0,0,1,1,0,1,0,0] => [1,1,0,0,1,1,0,1,0,0] => 1

[1,1,0,0,1,1,1,0,0,0] => [1,1,0,0,1,0,1,1,0,0] => 1

[1,1,0,1,0,0,1,0,1,0] => [1,1,0,1,0,1,1,0,0,0] => 0

[1,1,0,1,0,0,1,1,0,0] => [1,1,0,1,0,0,1,1,0,0] => 1

[1,1,0,1,0,1,0,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 0

[1,1,0,1,0,1,0,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => 1

[1,1,0,1,0,1,1,0,0,0] => [1,1,0,1,0,0,1,0,1,0] => 1

[1,1,0,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => 1

[1,1,0,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => 1

[1,1,0,1,1,0,1,0,0,0] => [1,1,0,0,1,1,0,0,1,0] => 2

[1,1,0,1,1,1,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0] => 1

[1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,0,1,1,0,0,0] => 1

[1,1,1,0,0,0,1,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => 1

[1,1,1,0,0,1,0,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => 1

[1,1,1,0,0,1,0,1,0,0] => [1,1,1,0,0,1,0,0,1,0] => 2

[1,1,1,0,0,1,1,0,0,0] => [1,1,1,0,0,0,1,0,1,0] => 1

[1,1,1,0,1,0,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => 1

[1,1,1,0,1,0,0,1,0,0] => [1,1,1,0,1,0,0,0,1,0] => 1

[1,1,1,0,1,0,1,0,0,0] => [1,0,1,1,0,1,0,0,1,0] => 1

[1,1,1,0,1,1,0,0,0,0] => [1,0,1,1,0,0,1,0,1,0] => 1

[1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => 1

[1,1,1,1,0,0,0,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => 1

[1,1,1,1,0,0,1,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => 1

[1,1,1,1,0,1,0,0,0,0] => [1,0,1,0,1,1,0,0,1,0] => 1

[1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 0

[1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => 0

[1,0,1,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,1,0,0,0,0,0] => 0

[1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => 0

[1,0,1,0,1,0,1,1,0,1,0,0] => [1,0,1,1,1,1,0,1,0,0,0,0] => 0

[1,0,1,0,1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,1,1,0,0,0,0] => 0

[1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,1,1,0,1,1,0,0,0,0,0] => 0

[1,0,1,0,1,1,0,0,1,1,0,0] => [1,0,1,1,1,0,1,1,0,0,0,0] => 0

[1,0,1,0,1,1,0,1,0,0,1,0] => [1,1,1,1,0,1,0,1,0,0,0,0] => 0

[1,0,1,0,1,1,0,1,0,1,0,0] => [1,0,1,1,1,0,1,0,1,0,0,0] => 0

[1,0,1,0,1,1,0,1,1,0,0,0] => [1,0,1,0,1,1,0,1,1,0,0,0] => 0

[1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => 1

[1,0,1,0,1,1,1,0,0,1,0,0] => [1,0,1,1,1,1,0,0,1,0,0,0] => 1

[1,0,1,0,1,1,1,0,1,0,0,0] => [1,0,1,0,1,1,1,0,1,0,0,0] => 0

[1,0,1,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,1,1,0,0,0] => 0

[1,0,1,1,0,0,1,0,1,0,1,0] => [1,1,1,0,1,1,1,0,0,0,0,0] => 0

[1,0,1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,0,1,1,1,0,0,0,0] => 0

[1,0,1,1,0,0,1,1,0,0,1,0] => [1,1,1,0,1,1,0,1,0,0,0,0] => 0

[1,0,1,1,0,0,1,1,0,1,0,0] => [1,0,1,1,0,1,1,0,1,0,0,0] => 0

[1,0,1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,0,0,1,1,1,0,0,0] => 1

[1,0,1,1,0,1,0,0,1,0,1,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => 0

[1,0,1,1,0,1,0,0,1,1,0,0] => [1,0,1,1,0,1,0,1,1,0,0,0] => 0

[1,0,1,1,0,1,0,1,0,0,1,0] => [1,1,1,0,1,0,1,0,1,0,0,0] => 0

[1,0,1,1,0,1,0,1,0,1,0,0] => [1,0,1,1,0,1,0,1,0,1,0,0] => 0

[1,0,1,1,0,1,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0,1,1,0,0] => 1

[1,0,1,1,0,1,1,0,0,0,1,0] => [1,1,1,0,1,1,0,0,1,0,0,0] => 1

[1,0,1,1,0,1,1,0,0,1,0,0] => [1,0,1,1,0,1,1,0,0,1,0,0] => 1

[1,0,1,1,0,1,1,0,1,0,0,0] => [1,0,1,1,0,0,1,1,0,1,0,0] => 1

[1,0,1,1,0,1,1,1,0,0,0,0] => [1,0,1,1,0,0,1,0,1,1,0,0] => 1

[1,0,1,1,1,0,0,0,1,0,1,0] => [1,1,1,1,0,0,1,1,0,0,0,0] => 1

[1,0,1,1,1,0,0,0,1,1,0,0] => [1,0,1,1,1,0,0,1,1,0,0,0] => 1

[1,0,1,1,1,0,0,1,0,0,1,0] => [1,1,1,1,0,0,1,0,1,0,0,0] => 1

[1,0,1,1,1,0,0,1,0,1,0,0] => [1,0,1,1,1,0,0,1,0,1,0,0] => 1

[1,0,1,1,1,0,0,1,1,0,0,0] => [1,0,1,1,1,0,0,0,1,1,0,0] => 1

[1,0,1,1,1,0,1,0,0,0,1,0] => [1,1,1,1,0,1,0,0,1,0,0,0] => 1

[1,0,1,1,1,0,1,0,0,1,0,0] => [1,0,1,1,1,0,1,0,0,1,0,0] => 1

[1,0,1,1,1,0,1,0,1,0,0,0] => [1,0,1,0,1,1,0,1,0,1,0,0] => 0

[1,0,1,1,1,0,1,1,0,0,0,0] => [1,0,1,0,1,1,0,0,1,1,0,0] => 1

>>> Load all 1513 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

Description

The number of factors DDU in a Dyck path.

Map

swap returns and last descent

Description

Return a Dyck path with number of returns and length of the last descent interchanged.
This is the specialisation of the map $\Phi$ in [1] to Dyck paths. It is characterised by the fact that the number of up steps before a down step that is neither a return nor part of the last descent is preserved.